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Level set methods for inverse scattering. (English) Zbl 1191.35272
Summary: We give an overview of recent techniques which use a level set representation of shapes for solving inverse scattering problems. The main focus is on electromagnetic scattering using different popular models, such as for example Maxwell’s equations, TM-polarized and TE-polarized waves, impedance tomography, a transport equation or its diffusion approximation. These models are also representative of a broader class of inverse problems. Starting out from the original binary approach of F. L. Santosa [ESAIM, Control Optim. Calc. Var. 1, 17–33 (1996; Zbl 0870.49016)] for solving the corresponding shape reconstruction problem, we successively develop more recent generalizations, such as for example using colour or vector level sets. Shape sensitivity analysis and topological derivatives are discussed as well in this framework. Moreover, various techniques for incorporating regularization into the shape inverse problem using level sets are demonstrated, which also include the choice of subclasses of simple shapes, such as ellipsoids, for the inversion. Finally, we present various numerical examples in two dimensions and in three dimensions for demonstrating the performance of level set techniques in realistic applications.

MSC:
35R30Inverse problems for PDE
35P25Scattering theory (PDE)
78A46Inverse scattering problems
49N45Inverse problems in calculus of variations
65J22Inverse problems (numerical methods in abstract spaces)
65N21Inverse problems (BVP of PDE, numerical methods)